This section is from the book "Turning And Mechanical Manipulation", by Charles Holtzapffel. Also available from Amazon: Turning and Mechanical Manipulation.

Every one in early life, has made the first step towards the acquirement of the various arts of working in sheet metal, in the simple process of making a box or tray of card; namely, by doubling up the four margins in succession to an equal width, then cutting out the small squares from the angles, and uniting the four sides of the box, either edge to edge, by paste, sealing-wax, or thread, or in similar manners by lapped or folded joints. A different mode is to make the sides of the box as a long strip, folded at all the angles but one; or lastly, the bottom and sides may be cut out entirely detached, and united in various ways.

In the above, and also in the most complicated vessels and solids it is necessary to depict on the material the exact shape of every plane superficies of the work, as in the plans and elevations of the architect; and these may be arranged in any clusters which admit of being folded together, so as to constitute part of the joints by bending the material. Thus, a hexagonal box, fig. 197, can be made by drawing first the hexagon required for the bottom, as in fig. 198, and erecting upon every side of the same a parallelogram equal to one of the sides, which in this case are all exactly alike; otherwise the group of sides can be drawn in a line, as in fig. 199, and bent upon the joints to the required angle, or 120 degrees. Either mode would be less troublesome than cutting out seven detached pieces and uniting them; the addition of one more hexagon, dotted in fig. 199, would serve to complete the top' of the hexagonal prism, by adding a cover or top surface.

The same mode will apply to polygonal figures of all kinds, regular or irregular; thus fig. 200 would be produced when the group of sides in 201 were bent around the irregular octagonal base; or that the sides of 202 were separately turned up.

The cylinder, is sometimes compared with a prism of so many sides, that they melt into each other and become a continuous curve; and if the hexagon in fig. 199 were replaced by a circle, and the group of sides were cut out of equal length with the circumference of that circle, and in width equal to the height of the vessel, any required cylinder could be produced. And in like manner any vessels of elliptical or similar forms, or those with parallel sides and curved ends, and all such combinations, could be made in the manner of fig. 201, (provided the sides were perpemlieular,) by cutting out a band equal in length to the collective margin of the figure, as measured by passing a string around it; or the sides might be made of two, or several pieces, if more convenient, or if requisite from their magnitude.

All prismatic vessels require parallelograms to be erected on their respective bases; but pyramids require triangles, and frustums of pyramids require trapezoids, as will be explained by figs. 201 and 205, which are the forms in which a single piece of metal must be cut, if required to produce fig. 203. Every one of the group of sides, must be indi\iduallv equal to one of the sides of the pyramid, whether it be regular or irregular, and 203 being an erect and equilateral figure, all the sides in 204 and 205 are required to be alike, and would be drawn from one templet: an irregular pyramid would require all its superficies to be drawn to their absolute forms and sizes.

The cone is sometimes compared with a pyramid with exceedingly numerous sides, (as the cylinder is compared with the prism,) and fig. 206, intended to make a funnel or the frustum of a cone of the same proportions as 203, illustrates this case. The sides of the cone are extended until they meet in the center o, fig. 203, and then with the slant distances o a, and o b, the two arcs, a a, and b b, are drawn with the compasses, from the center o; and so much of the arc a a, is required as equals the circumference a, of the cone: the margins a b, a b, are drawn as two radii. "When the figure is curled up until the radial sides meet, it will exactly equal the cone, and the similitude between figs. 205 and 206 is most explanatory, as 206 is just equal to the collective group of the sides required to form the pyramid.

It will now be easily seen that mixed polygonal figures, such as figs. 207, 209, and 211, may be produced in a similar manner, provided their sides are radiated from the square, the hexagonal or other bases, in the manner of figs. 208, 210, 212, but the sides of the rays not being straight, it is no longer possible to group them by their edges, as in figs. 199, 201, and 205. The object with plane surfaces, fig. 207, is only the meeting of two pyramids, at the ends of a prism, and when unfolded, as in fig. 208, the center a, is equal to the base a, of the object; the sides b, radiate and expand from the hexagon at the angle of the faces of the inverted or lower pyramid b, and their vertical heigh in the sheet is equal to the slant height in the vessel; the superficies c, are those of a prism, threfore they continue parallel, and have the vertical height of the part c, of the figure; lastly, the sides d, again contract as in the original, and at the same angle as the sides of the six upper faces; in a word, the faces b, c, d, are identical in the vase and in the radiated scheme.

Should the vessels, instead of planes, have surfaces of single curvature, as in figs. 209 and 211, the method is nearly as simple. The object is drawn on paper, and around its margin are marked several distances, either equal or unequal, and horizontal line-or ordinates are drawn from all to the central line. The radiating pieces for constructing the polygonal vases arc represented in figs. 210 and 212, in which the dotted lines are parallel with the sides of the hexagons or the bases, and at distances equal to those of the steps, 1, 2, 3, to 8, around the curve of the intended vases; the lengths of these lines, or ordinates, 1 1, 2 2, 3 3, are in legular hexagonal vessels exactly the same in the radiated plans as in the respective elevations, because the side of the hexagon and the radius of its circumscribing circle arc alike.

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